# Full text of "Treatise On Analysis Vol-Ii"

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```280      XIV   INTEGRATION IN LOCALLY COMPACT GROUPS

(14.8.1.1)   The image of the measure /x®£ under  the homeomorphism
(s, x)\-+(s, s~*x) ofG2 onto G2 is the measure \JL ® ft.

We may assume that n ^ 0. Then, for every F e j?T+(G2), the function
(s, x)\-+F(s, s'tx) belongs to jf+(G2), and we have

F(s, s~1x) dn(s) dft(x) =   dii(s)   F(s9 s~*x) dft(x)
JJ                                     J         J

(s9x)dff(x)

by virtue of the left-invariance of ft and the Lebesgue-Fubini theorem. This
proves the lemma.

Now suppose that p and /•/? are convolvable. Since |/*/?| = |/| •/?
(13.13.4) we may limit ourselves to the case where p. ^ 0 and/^ 0. For each
function he Jfc(G), the function (s9x)t-+h(sx)f(x) is (/x(g)j5)-integrable by
hypothesis (13.21.16 and 3.14.3); by virtue of (14.8.1.1), the same is true of the
function (jf9x)^h(x)f(s'1x) (13.7.10). If Afc is the set of points jceG such
that h(x) 7* 0, it follows from the theorem of Lebesgue-Fubini that there
exists a jS-negligible set NA in A/, such that, for each xeAHr\ ([Nft, the
function st-*f(s~1x) is /z-integrable. Taking a sequence of functions h in
JT+(G) such that the corresponding sets Ah cover G (4.5.2), we see that the
function st-*f(s~1x) is /^-integrable except at the points x of a ^-negligible set
N. Furthermore, it follows from the Lebesgue-Fubini theorem that the

function x\r-+h(x) \f(s~1x) dp,(s)9 defined almost everywhere with respect to
/?, is )S-integrable, and that

dp(x) = (h(x) dfi(x) tf(s-lx)

This proves that the condition of (14.8.1) is necessary, and that n * (/• ft) =
£•£(13.13.1).

Conversely, suppose that the condition is satisfied. We have to show that,
for each function /ze^+(G), the function (s,x)*~>h(sx)f(x) is(/*®/?)-
integrable. Now this function is (/x ® /?)-measurable because h is continuous
(13.21.13), and therefore it is enough to show that

JJsufficient that there should exist a ^-negligible set N such that the
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